Surprising features of the energy-mismatched nonlinear dimer.

The quantum mechanical energy-mismatched two-state system with cubic nonlinearity in its governing equation is surprisingly rich in its dynamics and has relevance to a number of subdisciplines of physics ranging from polaron phenomena to Bose-Einstein condensation. We review some of them that have been discussed recently and describe some new results that have not, pointing out their relevance in possible experiments.

A theory of coalescence of signaling receptor clusters in immune cells.

A theory of coalescence of signal receptor clusters in mast cells is developed in close connection with experiments. It is based on general considerations involving a feedback procedure and a time-dependent capture as part of a reaction-diffusion process. Characteristic features of observations that need to be explained are indicated and it is shown why calculations available in the literature are not satisfactory. While the latter involves static centers at which the reaction part of the phenomenon occurs, by its very nature, coalescence involves dynamically evolving centers. This is so because the process continuously modifies the size of the cluster aggregate which then proceeds to capture more material. We develop a procedure that consists of first solving a static reaction-diffusion problem and then imbuing the center with changing size. The consequence is a dependence of the size of the signal receptor cluster aggregate on time. A preliminary comparison with experiment is shown to reveal a sharp difference between theory and data. The observation indicates that the reaction occurs slowly at first and then picks up rapidly as time proceeds. Parameter modification to fit the observations cannot solve the problem. We use this observation to build into the theory an accumulation rate that is itself dependent on time. A memory representation and its physical basis are explained. The consequence is a theory that can be fit to observations successfully.

Analysis of Transmission of Infection in Epidemics: Confined Random Walkers in Dimensions Higher Than One.

The process of transmission of infection in epidemics is analyzed by studying a pair of random walkers, the motion of each of which in two dimensions is confined spatially by the action of a quadratic potential centered at different locations for the two walks. The walkers are animals such as rodents in considerations of the Hantavirus epidemic, infected or susceptible. In this reaction-diffusion study, the reaction is the transmission of infection, and the confining potential represents the tendency of the animals to stay in the neighborhood of their home range centers. Calculations are based on a recently developed formalism (Kenkre and Sugaya in Bull Math Biol 76:3016-3027, 2014) structured around analytic solutions of a Smoluchowski equation and one of its aims is the resolution of peculiar but well-known problems of reaction-diffusion theory in two dimensions. The resolution is essential to a realistic application to field observations because the terrain over which the rodents move is best represented as a 2-d landscape. In the present analysis, reaction occurs not at points but in spatial regions of dimensions larger than 0. The analysis uncovers interesting nonintuitive phenomena one of which is similar to that encountered in the one-dimensional analysis given in the quoted article, and another specific to the fact that the reaction region is spatially extended. The analysis additionally provides a realistic description of observations on animals transmitting infection while moving on what is effectively a two-dimensional landscape. Along with the general formalism and explicit one-dimensional analysis given in Kenkre and Sugaya (2014), the present work forms a model calculational tool for the analysis for the transmission of infection in dilute systems.

Analysis of Confined Random Walkers with Applications to Processes Occurring in Molecular Aggregates and Immunological Systems.

Explicit solutions are presented in the Laplace and time domains for a one-variable Fokker-Planck equation governing the probability density of a random walker moving in a confining potential. Illustrative applications are discussed in two unrelated physical contexts: quantum yields in a doped molecular crystal or photosynthetic system, and the motion of signal receptor clusters on the surface of a cell encountered in a problem in immunology. An interesting counterintuitive effect concerning the consequences of confinement is found in the former, and some insights into the driving force for microcluster centralization are gathered in the latter application.

Theory of the transmission of infection in the spread of epidemics: interacting random walkers with and without confinement.

A theory of the spread of epidemics is formulated on the basis of pairwise interactions in a dilute system of random walkers (infected and susceptible animals) moving in [Formula: see text] dimensions. The motion of an animal pair is taken to obey a Smoluchowski equation in [Formula: see text]-dimensional space that combines diffusion with confinement of each animal to its particular home range. An additional (reaction) term that comes into play when the animals are in close proximity describes the process of infection. Analytic solutions are obtained, confirmed by numerical procedures, and shown to predict a surprising effect of confinement. The effect is that infection spread has a non-monotonic dependence on the diffusion constant and/or the extent of the attachment of the animals to the home ranges. Optimum values of these parameters exist for any given distance between the attractive centers. Any change from those values, involving faster/slower diffusion or shallower/steeper confinement, hinders the transmission of infection. A physical explanation is provided by the theory. Reduction to the simpler case of no home ranges is demonstrated. Effective infection rates are calculated, and it is shown how to use them in complex systems consisting of dense populations.

Consequences of animal interactions on their dynamics: emergence of home ranges and territoriality.

Animal spacing has important implications for population abundance, species demography and the environment. Mechanisms underlying spatial segregation have their roots in the characteristics of the animals, their mutual interaction and their response, collective as well as individual, to environmental variables. This review describes how the combination of these factors shapes the patterns we observe and presents a practical, usable framework for the analysis of movement data in confined spaces. The basis of the framework is the theory of interacting random walks and the mathematical description of out-of-equilibrium systems. Although our focus is on modelling and interpreting animal home ranges and territories in vertebrates, we believe further studies on invertebrates may also help to answer questions and resolve unanswered puzzles that are still inaccessible to experimental investigation in vertebrate species.

Analytic solutions for some reaction-diffusion scenarios.

Motivated currently by the problem of coalescence of receptor clusters in mast cells in the general subject of immune reactions, and formerly by the investigation of exciton trapping and sensitized luminescence in molecular systems and aggregates, we present analytic expressions for survival probabilities of moving entities undergoing diffusion and reaction on encounter. Results we provide cover several novel situations in simple 1-d systems as well as higher-dimensional counterparts along with a useful compendium of such expressions in chemical physics and allied fields. We also emphasize the importance of the relationship of discrete sink term analysis to continuum boundary condition studies.

Quasi-one-dimensional waves in rodent populations in heterogeneous habitats: a consequence of elevational gradients on spatio-temporal dynamics.

Wave propagation can be clearly discerned in data collected on mouse populations in the Cibola National Forest (New Mexico, USA) related to seasonal changes. During an exploration of the construction of a methodology for investigations of the spread of the Hantavirus epidemic in mice we have built a system of interacting reaction diffusion equations of the Fisher-Kolmogorov-Petrovskii-Piskunov type. Although that approach has met with clear success recently in explaining Hantavirus refugia and other spatiotemporal correlations, we have discovered that certain observed features of the wave propagation observed in the data we mention are impossible to explain unless modifications are made. However, we have found that it is possible to provide a tentative explanation/description of the observations on the basis of an assumed Allee effect proposed to exist in the dynamics. Such incorporation of the Allee effect has been found useful in several of our recent investigations both of population dynamics and pattern formation and appears to be natural to the observed system. We report on our investigation of the observations with our extended theory.

Reaction-diffusion theory in the presence of an attractive harmonic potential.

Problems involving the capture of a moving entity by a trap occur in a variety of physical situations, the moving entity being an electron, an excitation, an atom, a molecule, a biological object such as a receptor cluster, a cell, or even an animal such as a mouse carrying an epidemic. Theoretical considerations have almost always assumed that the particle motion is translationally invariant. We study here the case when that assumption is relaxed, in that the particle is additionally subjected to a harmonic potential. This tethering to a center modifies the reaction-diffusion phenomenon. Using a Smoluchowski equation to describe the system, we carry out a study which is explicit in one dimension but can be easily extended for arbitrary dimensions. Interesting features emerge depending on the relative location of the trap, the attractive center, and the initial placement of the diffusing particle.

Extinction of refugia of hantavirus infection in a spatially heterogeneous environment.

We predict an abrupt observable transition, on the basis of numerical studies, of hantavirus infection in terrain characterized by spatially dependent environmental resources. The underlying framework of the analysis is that of Fisher equations with an internal degree of freedom, the state of infection. The unexpected prediction is of the sudden disappearance of refugia of infection in spite of the existence of supercritical (favorable) food resources, brought about by reduction of their spatial extent. Numerical results are presented and a theoretical explanation is provided on analytic grounds on the basis of the competition of diffusion of rodents carrying the hantavirus and nonlinearity present in the resource interactions.

Analytical studies of fronts, colonies, and patterns: Combination of the Allee effect and nonlocal competition interactions.

We present an analytic study of traveling fronts, localized colonies, and extended patterns arising from a reaction-diffusion equation which incorporates simultaneously two features: the well-known Allee effect and spatially nonlocal competition interactions. The former is an essential ingredient of most systems in population dynamics and involves extinction at low densities, growth at higher densities, and saturation at still higher densities. The latter feature is also highly relevant, particularly to biological systems, and goes beyond the unrealistic assumption of zero-range interactions. We show via exact analytic methods that the combination of the two features yields a rich diversity of phenomena and permits an understanding of a variety of issues including spontaneous appearance of colonies.

Effects of rotation on the nonlinear friction of a damped dimer sliding on a periodic substrate.

Rotational effects on the nonlinear sliding friction of a damped dimer moving over a substrate are studied within a largely one-dimensional model. The model consists of two masses connected rigidly, internally damped, and sliding over a sinusoidal (substrate) potential while being free to rotate in the plane containing the masses and the direction of sliding. Numerical simulations of the dynamics performed by throwing the dimer with an initial center of mass velocity along the substrate direction show a richness of phenomena including the appearance of three separate regimes of motion. The orientation of the dimer performs tiny oscillations around values that are essentially constant in each regime. The constant orientations form an intricate pattern determined by the ratio of the dimer length to the substrate wavelength as well as by the initial orientations chosen. Corresponding evolution of the center of mass velocity consists, respectively, of regular oscillations in the first and the third regimes, but a power law decay in the second regime; the center of mass motion is effectively damped in this regime because of the coupling to the rotation. Depending on the initial orientation of the dimer, there is considerable variation in the overall behavior. For small initial angles to the vertical, an interesting formal connection can be established to earlier results known in the literature for a vibrating, rather than rotating, dimer. But for large angles, on which we focus in the present paper, quite different evolution occurs. Some of the numerical observations are explained successfully on the basis of approximate analytical arguments but others pose puzzling problems.

Theory of possible effects of the Allee phenomenon on the population of an epidemic reservoir.

We investigate possible effects of high-order nonlinearities on the shapes of infection refugia of the reservoir of an infectious disease. We replace Fisher-type equations that have been recently used to describe, among others, the Hantavirus spread in mouse populations by generalizations capable of describing Allee effects that are a consequence of the high-order nonlinearities. After analyzing the equations to calculate steady-state solutions, we study the stability of those solutions and compare to the earlier Fisher-type case. Finally, we consider the spatial modulation of the environment and find that unexpected results appear, including a bifurcation that has not been studied before.

Phase transitions induced by complex nonlinear noise in a system of self-propelled agents.

We propose a comprehensive dynamical model for cooperative motion of self-propelled particles, e.g., flocking, by combining well-known elements such as velocity-alignment interactions, spatial interactions, and angular noise into a unified Lagrangian treatment. Noise enters into our model in an especially realistic way: it incorporates correlations, is highly nonlinear, and it leads to a unique collective behavior. Our results show distinct stability regions and an apparent change in the nature of one class of noise-induced phase transitions, with respect to the mean velocity of the group, as the range of the velocity-alignment interaction increases. This phase-transition change comes accompanied with drastic modifications of the microscopic dynamics, from nonintermittent to intermittent. Our results facilitate the understanding of the origin of the phase transitions present in other treatments.

Extensions of effective-medium theory of transport in disordered systems.

The effective-medium theory of transport in disordered systems, whose basis is the replacement of spatial disorder by temporal memory, is extended in several practical directions. Restricting attention to a one-dimensional system with bond disorder for specificity, a transformation procedure is developed to deduce explicit expressions for the memory functions from given distribution functions characterizing the system disorder. It is shown how to use the memory functions in the Laplace domain forms in which they first appear, and in the time domain forms which are obtained via numerical inversion algorithms, to address time evolution of the system beyond the asymptotic domain of large times normally treated. An analytic but approximate procedure is provided to obtain the memories, in addition to the inversion algorithm. Good agreement of effective-medium theory predictions with numerically computed exact results is found for all time ranges for the distributions used except near the percolation limit, as expected. The use of ensemble averages is studied for normal as well as correlation observables. The effect of size on effective-medium theory is explored and it is shown that, even in the asymptotic limit, finite-size corrections develop to the well-known harmonic mean prescription for finding the effective rate. A percolation threshold is shown to arise even in one dimension for finite (but not infinite) systems at a concentration of broken bonds related to the system size. Spatially long-range transfer rates are shown to emerge naturally as a consequence of the replacement of spatial disorder by temporal memories, in spite of the fact that the original rates possess nearest neighbor character. Pausing time distributions in continuous-time random walks corresponding to the effective-medium memories are calculated.

Multifractality of random walks in the theory of vehicular traffic.

We investigate the origin of the experimentally observed multifractal scaling of vehicular traffic flows by studying a hydrodynamic model of traffic. We first extend and apply the formalism of generalized Hurst exponents H(q) to the case of random walkers that not only diffuse but rather also undergo nonlinear convection due to interactions with other walkers. We recover analytically, as expected, that H(q) equals 12 for a single random walker starting at the origin whose probability density function satisfies Burger's equation. Despite this result for a single walker, we find that for a collection of nonlinearly convecting diffusive particles, transient effects can give rise to multiscaling at given time scales for many initial conditions. In the Lighthill-Whitham-Richards hydrodynamic model of traffic, this multiscaling effect becomes more prominent for smaller diffusion constants and larger speed limits. We discuss the relevance of these findings for the realistic scenario of traffic that flows from small roads to large highways and vice versa, where transient effects can be expected to play a significant role.

Nonlinearity in bacterial population dynamics: proposal for experiments for the observation of abrupt transitions in patches.

An explicit proposal for experiments leading to abrupt transitions in spatially extended bacterial populations in a Petri dish is presented on the basis of an exact formula obtained through an analytic theory. The theory provides accurately the transition expressions despite the fact that the actual solutions, which involve strong nonlinearity, are inaccessible to it. The analytic expressions are verified through numerical solutions of the relevant nonlinear equation. The experimental setup suggested uses opaque masks in a Petri dish bathed in ultraviolet radiation [Lin A-L, et al. (2004) Biophys J 87:75-80 and Perry N (2005) J R Soc Interface 2:379-387], but is based on the interplay of two distances the bacteria must traverse, one of them favorable and the other adverse. As a result of this interplay feature, the experiments proposed introduce highly enhanced reliability in interpretation of observations and in the potential for extraction of system parameters.

Molecular motion in cell membranes: analytic study of fence-hindered random walks.

A theoretical calculation is presented to describe the confined motion of transmembrane molecules in cell membranes. The study is analytic, based on Master equations for the probability of the molecules moving as random walkers, and leads to explicit usable solutions including expressions for the molecular mean square displacement and effective diffusion constants. One outcome is a detailed understanding of the dependence of the time variation of the mean square displacement on the initial placement of the molecule within the confined region. How to use the calculations is illustrated by extracting (confinement) compartment sizes from experimentally reported published observations from single particle tracking experiments on the diffusion of gold-tagged G -protein coupled mu -opioid receptors in the normal rat kidney cell membrane, and by further comparing the analytical results to observations on the diffusion of phospholipids, also in normal rat kidney cells.

Random-walk access times on partially disordered complex networks: an effective medium theory.

An analytic effective medium theory is constructed to study the mean access times for random walks on hybrid disordered structures formed by embedding complex networks into regular lattices, considering transition rates F that are different for steps across lattice bonds from the rates f across network shortcuts. The theory is developed for structures with arbitrary shortcut distributions and applied to a class of partially disordered traversal enhanced networks in which shortcuts of fixed length are distributed randomly with finite probability. Numerical simulations are found to be in excellent agreement with predictions of the effective medium theory on all aspects addressed by the latter. Access times for random walks on these partially disordered structures are compared to those on small-world networks, which on average appear to provide the most effective means of decreasing access times uniformly across the network.

Effect of predators of juvenile rodents on the spread of the hantavirus epidemic.

Effects of predators of juvenile mice on the spread of the Hantavirus are analyzed in the context of a recently proposed model. Two critical values of the predation probability are identified. When the smaller of them is exceeded, the hantavirus infection vanishes without extinguishing the mice population. When the larger is exceeded, the entire mice population vanishes. These results suggest the possibility of control of the spread of the epidemic by introducing predators in areas of mice colonies in a suitable way so that such control does not kill all the mice but lowers the epidemic spread.